What is an exponent and how do I calculate it?

An exponent represents repeated multiplication of a base number. In x^y, x is the base and y is the exponent (or power). For example, 2^3 means 2 × 2 × 2 = 8. To calculate exponents: for positive integers, multiply the base by itself y times; for negative exponents like 2^(-3), use the reciprocal: 1/2^3 = 1/8 = 0.125; for fractional exponents like 8^(1/3), take the root: ³√8 = 2. For zero exponents, any non-zero number raised to 0 equals 1 (x^0 = 1). Our calculator handles all types of exponents including decimals, negatives, and fractions, showing step-by-step solutions for each calculation.

What are the main exponent rules?

The main exponent rules are: 1) Product Rule: x^a × x^b = x^(a+b) - when multiplying same bases, add exponents. 2) Quotient Rule: x^a ÷ x^b = x^(a-b) - when dividing same bases, subtract exponents. 3) Power Rule: (x^a)^b = x^(a×b) - power of a power means multiply exponents. 4) Zero Exponent: x^0 = 1 - any non-zero number to power 0 equals 1. 5) Negative Exponent: x^(-a) = 1/x^a - negative means reciprocal. 6) Fractional Exponent: x^(1/n) = ⁿ√x - fractional means root. 7) Power of Product: (xy)^a = x^a × y^a. 8) Power of Quotient: (x/y)^a = x^a / y^a. These rules are fundamental in algebra and are displayed with examples in our calculator.

How do I calculate exponential growth and decay?

Exponential growth and decay follow the formula A = P(1 + r/n)^(nt), where P is the initial value, r is the rate (as a decimal), n is the number of compounding periods per time unit, and t is time. For growth, r is positive (e.g., 5% = 0.05); for decay, r is negative (e.g., -3% = -0.03). Example: $1000 growing at 5% annually for 10 years with annual compounding (n=1): A = 1000(1 + 0.05/1)^(1×10) = 1000(1.05)^10 = $1628.89. For monthly compounding (n=12), use A = 1000(1 + 0.05/12)^(12×10) = $1647.01. Exponential growth is common in population studies, compound interest, and bacterial growth. Decay applies to radioactive decay, depreciation, and cooling.

What is the difference between exponential and linear growth?

Linear growth increases by a constant amount each period (y = mx + b), while exponential growth increases by a constant percentage (y = a × b^x). For example, adding $100 each year is linear: Year 1: $100, Year 2: $200, Year 3: $300. Growing by 10% each year is exponential: Year 1: $100, Year 2: $110, Year 3: $121. Exponential growth accelerates over time and eventually grows much faster than linear growth. Real-world exponential examples include compound interest, viral spread, and population growth. Linear examples include saving a fixed amount monthly or constant speed travel. Exponential growth shows a characteristic J-curve when graphed, while linear growth shows a straight line.

How do negative and fractional exponents work?

Negative exponents indicate reciprocals: x^(-n) = 1/x^n. Example: 5^(-2) = 1/5^2 = 1/25 = 0.04. This makes sense because x^a ÷ x^b = x^(a-b), so x^2 ÷ x^5 = x^(2-5) = x^(-3) = 1/x^3. Fractional exponents indicate roots: x^(1/n) means the nth root of x. Example: 16^(1/2) = √16 = 4, and 8^(1/3) = ³√8 = 2. For x^(m/n), take the nth root then raise to the mth power: 8^(2/3) = (³√8)^2 = 2^2 = 4. This connects exponents and radicals: √x = x^(1/2), ³√x = x^(1/3). Negative fractional exponents combine both: 16^(-1/2) = 1/16^(1/2) = 1/4 = 0.25.

Exponent Calculator - Calculate Powers, Growth & Exponential Functions | Free Tool

Base (x) *

The number being raised to a power

Exponent (y) *

The power to which the base is raised

Examples:

• Positive exponent: 2^3 = 8
• Negative exponent: 2^(-2) = 0.25
• Fractional exponent: 8^(1/3) = 2
• Zero exponent: 5^0 = 1

Power Calculation Result

Enter values and click Calculate to see results

Exponent Rules Reference

Product Rule

x^a × x^b = x^(a+b)

When multiplying same bases, add exponents

Example:

2^3 × 2^4 = 2^(3+4) = 2^7 = 128

Quotient Rule

x^a ÷ x^b = x^(a-b)

When dividing same bases, subtract exponents

Example:

5^6 ÷ 5^2 = 5^(6-2) = 5^4 = 625

Power Rule

(x^a)^b = x^(a×b)

Power of a power: multiply exponents

Example:

(3^2)^3 = 3^(2×3) = 3^6 = 729

Zero Exponent

x^0 = 1

Any non-zero number to power 0 equals 1

Example:

999^0 = 1
(-5)^0 = 1

Negative Exponent

x^(-a) = 1 / x^a

Negative exponent means reciprocal

Example:

2^(-3) = 1/2^3 = 1/8 = 0.125

Fractional Exponent

x^(1/n) = ⁿ√x

Fractional exponent means root

Example:

16^(1/2) = √16 = 4
8^(1/3) = ³√8 = 2

Power of Product

(xy)^a = x^a × y^a

Distribute exponent to each factor

Example:

(2×3)^2 = 2^2 × 3^2 = 4 × 9 = 36

Power of Quotient

(x/y)^a = x^a / y^a

Distribute exponent to numerator and denominator

Example:

(6/3)^2 = 6^2 / 3^2 = 36/9 = 4

Understanding Exponents

An exponent (also called a power or index) indicates how many times a number (the base) is multiplied by itself. In the expression x^y, x is the base and y is the exponent.

Basic Examples:

• 2^3 = 2 × 2 × 2 = 8
• 5^2 = 5 × 5 = 25
• 10^4 = 10 × 10 × 10 × 10 = 10,000
• 3^1 = 3 (any number to power 1 is itself)
• 7^0 = 1 (any non-zero number to power 0 is 1)

Types of Exponents

Positive Integer Exponents

Represent repeated multiplication.

4^3 = 4 × 4 × 4 = 64

Negative Exponents

Represent reciprocals (1 divided by the positive power).

2^(-3) = 1/2^3 = 1/8 = 0.125

Fractional Exponents

Represent roots (1/n means nth root).

16^(1/2) = √16 = 4
8^(1/3) = ³√8 = 2

Zero Exponent

Any non-zero number to power 0 equals 1.

999^0 = 1
(-5)^0 = 1

Exponent Rules Explained

Product Rule

When multiplying powers with the same base, add the exponents:

x^a × x^b = x^(a+b)
Example: 3^2 × 3^4 = 3^(2+4) = 3^6 = 729

Quotient Rule

When dividing powers with the same base, subtract the exponents:

x^a ÷ x^b = x^(a-b)
Example: 5^7 ÷ 5^3 = 5^(7-3) = 5^4 = 625

Power Rule

When raising a power to another power, multiply the exponents:

(x^a)^b = x^(a×b)
Example: (2^3)^2 = 2^(3×2) = 2^6 = 64

Power of a Product

Distribute the exponent to each factor in the product:

(xy)^a = x^a × y^a
Example: (2 × 3)^2 = 2^2 × 3^2 = 4 × 9 = 36

Exponential Growth and Decay

Exponential growth occurs when a quantity increases by a constant percentage over equal time periods. Exponential decay is the opposite - decreasing by a constant percentage.

The Compound Interest Formula:

A = P(1 + r/n)^(nt)

• A = Final amount
• P = Initial principal (starting value)
• r = Annual interest rate (as a decimal)
• n = Number of times compounded per year
• t = Time in years

Growth Example

$1000 at 5% annual interest for 10 years:

A = 1000(1 + 0.05/1)^(1×10)
A = 1000(1.05)^10
A = $1,628.89

Decay Example

$1000 depreciating 5% annually for 10 years:

A = 1000(1 - 0.05/1)^(1×10)
A = 1000(0.95)^10
A = $598.74

Real-World Applications

💰 Finance

• Compound interest
• Investment growth
• Loan calculations
• Retirement planning

🔬 Science

• Bacterial growth
• Radioactive decay
• Population dynamics
• Chemical reactions

💻 Technology

• Computer algorithms
• Data compression
• Viral marketing
• Network effects

Related Calculators

🔬

Power Calculation Result

Exponent Rules Reference

Product Rule

Quotient Rule

Power Rule

Zero Exponent

Negative Exponent

Fractional Exponent

Power of Product

Power of Quotient

Understanding Exponents

Types of Exponents

Positive Integer Exponents

Negative Exponents

Fractional Exponents

Zero Exponent

Exponent Rules Explained

Product Rule

Quotient Rule

Power Rule

Power of a Product

Exponential Growth and Decay

The Compound Interest Formula:

Growth Example

Decay Example

Real-World Applications

💰 Finance

🔬 Science

💻 Technology

Related Calculators

Scientific Notation

Square Root

Percentage Calculator

Exponent Calculator - Calculate Powers, Exponential Growth & Exponent Rules

Power Calculation Result

Exponent Rules Reference

Product Rule

Quotient Rule

Power Rule

Zero Exponent

Negative Exponent

Fractional Exponent

Power of Product

Power of Quotient

Understanding Exponents

Types of Exponents

Positive Integer Exponents

Negative Exponents

Fractional Exponents

Zero Exponent

Exponent Rules Explained

Product Rule

Quotient Rule

Power Rule

Power of a Product

Exponential Growth and Decay

The Compound Interest Formula:

Growth Example

Decay Example

Real-World Applications

💰 Finance

🔬 Science

💻 Technology

Related Calculators

Scientific Notation

Square Root

Percentage Calculator